Later on, a lot of attention has been paid to such type of solutions for different nonlinear transport fluid models like Euler equations in the plane or in the unit disc, generalised surface quasi-geostrophic equations \((SQG)_{\alpha }\)  and \((QGSW)_{\lambda }\)  equations. Also, several topological and regularity settings were explored and we may refer to
[1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}. Nevertheless, it seems that the Euler-\(\alpha \)  model has been set aside from these studies. Hence, we propose here to study the emergence of simply-connected V-states for this model. For this task, due to the decomposition (REF ), we may emphasize the result in [2]}, where they found the following angular velocities related to the modified Bessel functions for which the bifurcation of simply-connected V-states from the unit disc occurs
\(\Omega _{\mathbf {m}}^{\textnormal {\tiny {SW}}}(\lambda )\triangleq I_1(\lambda )K_1(\lambda )-I_{\mathbf {m}}(\lambda )K_{\mathbf {m}}(\lambda ).\) 
